The partition function of 2d conformal field theory is a modular invariant function. It is known that the partition function of a holomorphic CFT whose central charge is a multiple of 24 is a polynomial in the Klein function. In this paper, by using the medium temperature expansion we show that every modular invariant partition function can be mapped to a holomorphic partition function whose structure can be determined similarly. We use this map to study partition function of CFTs with half-integer left and right conformal weights. We show that the corresponding left and right central charges are necessarily multiples of 4. Furthermore, the degree of degeneracy of high-energy levels can be uniquely determined in terms of the degeneracy in the low energy states.

Using modular bootstrap we show the lightest primary fields of a unitary compact two dimensional conformal field theory(with c,c > 1) has a conformal weight h 1 ≤ c 12 + O(1).This implies that the upper bound on the dimension of the lightest primary fields depends on their spin. In particular if the set of lightest primary fields includes extremal or near extremal states whose spin to dimension ratio j ∆ ≈ 1, the corresponding dimension is ∆ ≤ c 12 + O(1). From AdS/CFT correspondence, we obtain an upper bound on the spectrum of black hole in three dimensional gravity. Our results show that if the first primary fields have large spin, the corresponding three dimensional gravity has extremal or near extremal BTZ black hole. * E-mail address: mashrafi@ucdavis.edu 1 For recent development in modular bootstrap see [1]- [8]

In this paper, we study mathematical functions of relevance to pure gravity in AdS3. Modular covariance places stringent constraints on the space of such functions; modular invariance places even stronger constraints on how they may be combined into physically viable candidate partition functions. We explicitly detail the list of holomorphic and anti-holomorphic functions that serve as candidates for chiral and anti-chiral partition functions and note that modular covariance is only consistent with such functions when the left (resp. right) central charge is an integer multiple of 8, c ∈ 8ℕ. We then find related constraints on the symmetry group of the corresponding topological, Chern-Simons, theory in the bulk of AdS. The symmetry group of the theory can be one of two choices: either SO(2; 1) × SO(2; 1) or its three-fold diagonal cover. We introduce the generalized Hecke operators which map the modular covariant functions to the modular covariant functions. With these mathematical results, we obtain conjectural partition functions for extremal CFT2s, and the corresponding microcanonical entropies, when the chiral central charges are multiples of eight. Finally, we compute subleading corrections to the Beckenstein-Hawking entropy in the bulk gravitational theory with these conjectural partition functions.